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Theorem rspcegf 41157
Description: A version of rspcev 3620 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rspcegf.1 𝑥𝜓
rspcegf.2 𝑥𝐴
rspcegf.3 𝑥𝐵
rspcegf.4 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspcegf ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Proof of Theorem rspcegf
StepHypRef Expression
1 rspcegf.2 . . . 4 𝑥𝐴
2 rspcegf.3 . . . . . 6 𝑥𝐵
31, 2nfel 2989 . . . . 5 𝑥 𝐴𝐵
4 rspcegf.1 . . . . 5 𝑥𝜓
53, 4nfan 1891 . . . 4 𝑥(𝐴𝐵𝜓)
6 eleq1 2897 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 rspcegf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7anbi12d 630 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
91, 5, 8spcegf 3588 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
109anabsi5 665 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
11 df-rex 3141 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
1210, 11sylibr 235 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wnf 1775  wcel 2105  wnfc 2958  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494
This theorem is referenced by:  rspcef  41211  stoweidlem46  42208
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